Surface roughness characterization using representative elementary area (REA) analysis

We proposed the Representative Elementary Area (REA) analysis method and illustrated how it is needed to evaluate representative roughness parameters of surfaces. We used mean height (Sa) roughness to study how its variations converge to a steady state as we expanded the area of investigation (AOI) using combined scan tiles obtained through Confocal Laser Scanning Microscopy. We tested quartz and glass surfaces, subjecting them to various levels of polishing with grit sizes ranging between # 60 and #1200. The scan tiles revealed a multiscale roughness texture characterized by the dominance of valleys over peaks, lacking a fractal nature. REA analysis revealed Sa variations converged to a steady state as AOI increased, highlighting the necessity of the proposed method. The steady-state Sa, denoted as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\text{Sa}}}_{{\text{REA}}}$$\end{document}SaREA, followed an inverse power law with polishing grit size, with its exponent dependent on the material hardness. The REA length representing \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\text{Sa}}}_{{\text{REA}}}$$\end{document}SaREA of glass surfaces, followed another inverse power law with polishing grit size and an indeterminate relationship for quartz surfaces. The multiscale characteristics and convergence to steady state were also evident in skewness, kurtosis, and autocorrelation length (Sal) parameters. Sal increased to a maximum value before decreasing linearly as AOI was linearly increased. The maximum Sal, termed as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\text{Sal}}}_{{\text{max}}}$$\end{document}Salmax, exhibited a linear relationship with REA. In the absence of REA analysis, the magnitude of uncertainty depended on the polishing grit size. Finely polished surfaces exhibited a 10–20% variability, which increased to up to 70% relative to the steady-state Sa with coarser polishing.

Parametric methods provide single values for parameters, e.g., RMS or mean height (Sa), skewness, and kurtosis over a finite surface.Functional methods involve determining the autocorrelation function (ACF), correlation length, and power spectral density (PSD), allowing roughness characterization across a wide range of wavelengths within subsets of a finite surface (Table 1).Fractal characterization is based on the premise that roughness asperities are nested within smaller asperities, creating a hierarchical structure 12,32,38 .Directional characterization aids in understanding the anisotropy of roughness in different directions, which is relevant to surfaces with preferred orientations, such as faults or joints.
The dependence of roughness parameters on the length scale renders statistical parameters insufficient for determining representative roughness characteristics 32 unless their scale independence is established 25 .ACF and correlation length 25,33 indicate the scale at which statistical parameters become scale-invariant.However, correlation length has been shown to increase with sampling intervals or length scales 25,39,40 , exhibiting uncertainty in their use to determine scale-invariant roughness parameters.The Fast Fourier Transform (FFT) of ACF gives the Power Spectrum Density (PSD) enabling the evaluation of waviness and unevenness, including long and short wavelength roughness of surfaces 28 .While PSD is useful for characterizing surface topography with shorter wavelength roughness 12 , it faces challenges in detecting large wavelength characteristics 28 .In contrast, longer wavelengths significantly impact surface roughness 30 .Additionally, the reliable calculation of PSD 38 and its application to non-stationary surfaces is challenging 28,41 , including the determination of the sampling window 26 for non-periodic surfaces 38 .
In contrast, fractal analysis of surface roughness is desired 41 because it offers a scale-independent characterization 32 , eliminating concerns related to scan size or resolution 38 .Fractal analysis aims to establish a power-law relationship, such as between RMS height and the sampling window or scan size 12 .This relationship is characterized by an exponent known as the Hurst exponent 28 .However, surfaces are known to demonstrate the scale-invariant property over a limited range of length scales in practice 12,41 .Gujrati et al. illustrated a powerlaw fractal relationship at smaller length scales or scan size, whereas Fardin et al., 37 showed the Hurst exponent asymptotes to a constant at larger length scales.The power-law fractal relationship tends to a 'roll-off ' to a constant value beyond a certain larger wavelength 35 or scan size 12 .
We postulate that the minimum length scale or wavelength at which the power-law relationship rolls off to a constant represents the threshold for determining statistical roughness parameters, such as RMS height or mean height.However, not all surfaces, particularly non-natural surfaces, exhibit fractal characteristics, leading to ambiguity in determining representative surface roughness parameters.In the absence of systematic testing for length scale independence of the roughness parameter, errors can range up to 30% 25 or even exceed an order of magnitude 12 .
When the representative surface roughness parameter is unknown, conflicting relationships can emerge between the roughness parameter and related phenomena such as adhesion, wettability, friction, and hydromechanical response 12,17,28,40,42 .For example, contradictory experimental data regarding the roughness parameter and contact angle have been reported for both non-geologic material surfaces 17,42,43 and geologic mineral surfaces 16,17  The objective of this study is to determine a length-scale-independent representative surface roughness parameter.To achieve this, we polish quartz and glass surfaces with six different grit sizes from coarse to fine, following a thin-section preparation routine.We scan the surfaces using CSLM at a magnification (i.e., with a 50 × objective lens) beyond which no discernable changes in height distribution are found.A tile-scan mode is used to linearly extend 2D surface scans up to 2500 µm.We analyze statistical parameters (mean height, skewness, and kurtosis), and employ functional methods (autocorrelation function and autocorrelation length) while enlarging the AOI across the extended 2D scan tiles.We apply principles of continuum mechanics to investigate how roughness parameters evolve until they converge to a steady state.The finite area at which this steady state is observed is referred to as the representative elementary area (REA).

Single tile scan of smoother surfaces and REA
To compare surface texture variation and demonstrate the length-scale effect on surface roughness evaluation, we begin using single tile scans, i.e., field of view (FOV) of 196 µm × 254 µm) of the finest polished quartz and glass, frosted glass, and clear glass surfaces.The results obtained from these single-tile scans highlight the potential for misinterpreting roughness parameters and emphasize the necessity of conducting REA analysis on larger areas.Engineered clear and frosted glass surfaces were included in this analysis, as they were expected to exhibit no length scale dependence in roughness parameters.We compared these surfaces to the finest polished quartz and glass (grit #1200), anticipating similar length scale independence.
The polished quartz and glass surfaces exhibit comparable levels of asperities but differ in spatial pattern due to the two different polishing routines discussed in the methods section.Figure 1 provides a comparison of asperities between polished quartz and glass, frosted glass surfaces using a single scale bar.A unique scale, however, is required for the clear glass surface to reveal its fine-scale asperities (Fig. 1d).
All single-tile surfaces exhibit randomly distributed asperities (Fig. 1), implying that a significantly smaller area than the single-tile area is sufficient to evaluate a representative roughness parameter.The mean height of these asperities, however, shows a clear length scale dependence (Fig. 2).The mean height (Sa) is the average of the absolute height values (z) at locations (x, y) within the specified evaluation area or AOI, calculated in accordance with ISO 25178 1 as: where A is the sampling area, and z x, y is the ordinate or height at a given (x, y).We assess the length scale dependence of the mean height roughness parameter (Sa) by incrementally expanding the sample window or AOI along the x-direction, starting at x = 0.The size of the sample window in the y-direction is kept constant so that an increase in the sample window is the same as an increase in length, L (Fig. 1a).
Engineered clear and frosted glass surfaces display variation in Sa with sample length until reaching L ~ 125 (µm), after which Sa attains a steady state.We observed no evidence of fractal or power-law scaling between Sa and L. The steady-state values are highlighted by the red dashed line in Fig. 2. The L ~ 125 (µm), marked by red diamonds (Fig. 2), represents the minimum sample length or REA required to assess the representative mean height (Sa) roughness, even for the engineered clear glass surface.Notably, this sample length, L ~ 125 (µm), significantly exceeds the capabilities of methods like AFM.
In the case of polished quartz and glass surfaces (i.e., grit #1200), Sa also exhibits variations with sample length, which converge to a steady state beyond L ~ 125 µm.However, these steady values of polished surfaces are a false account of the representative Sa since larger wavelength asperities have yet to be included in determining the REA from linearly combined scan tiles, as discussed in section "REA analysis using mean height (Sa)".

Surface texture from linear tile scans
We utilized the tile-scan mode of CLSM to obtain large scan areas for determining representative roughness parameters of polished surfaces that remain steady as the length or AOI increases.Up to ten single tile scans were linearly combined in the x-direction with a 10% overlap to reach a total length of 2500 µm.Since the polishing was uniform in all directions, a linear extension of the scan area was expected to be sufficient to evaluate representative surface roughness characteristics.However, for visual clarity, Figs. 3 and 4 display six combined tiles.
The combined quartz surfaces, polished with a sequential polishing method, revealed a repeating nature of larger wavelength asperities, illustrating the multiscale nature of surface roughness (Fig. 2).The magnitude of these asperities decreased with increasing polishing fineness, from grit #60-#1200.Notably, coarse quartz polishing (grit #60) exhibited roughly equal proportions of peaks and valleys.Subsequent polishing steps reduced the magnitude of peaks and their repeating frequency; for example, sequential polishing to grit #600 almost eliminated peaks, and sequential polishing to grit #320 reduced the repeating nature of peaks (Fig. 3e,f).The characteristics of valleys, however, persisted in the surface texture evolution during polishing.
Similarly, the combined glass surfaces, polished with an individual polishing method, also revealed repeating large-wavelength asperities (Fig. 4), with the magnitude of peaks and valleys being approximately half that of quartz surfaces.Polishing glass surfaces with grit # ≤ 320 resulted in asperities with regions of peaks and valleys spanning longer lengths, while grit # ≥ 600 led to random variations in peaks and valleys, indicating the absence of multiscale roughness (Fig. 4e,f).Additionally, all individually polished glass samples showed a similar distribution of peaks and valleys, highlighting variations in surface texture development compared to quartz samples, influenced by sample crystallinity, hardness, and polishing method.
Lastly, we presented 1D height profiles from combined scan tiles for comparative analysis (Fig. 5).While recognizing the limitations of using 1D profiles for roughness parameter determination 34 , we explored whether (1) Sa = 1 A z x, y dxdy the estimated 1D mean height (Ra) roughness parameter from significantly longer profiles aligned with Sa from 2D surfaces.We examined three 1D profile sections from the bottom, middle, and top locations of 2D surfaces.The 1D profiles of quartz samples from the middle section illustrated how sequential polishing diminished the magnitude and frequency of peaks and valleys, with differences observed up to grit #320 (Fig. 5a-f).In contrast, the 1D profiles from glass samples exhibited shorter and longer length variations (Fig. 5g-l), highlighting material-specific differences in surface texture development between the two materials.We further used mean height roughness determined from these 1D profiles to evaluate the uncertainty presented in section "Sa uncertainty in the absence of REA analysis".www.nature.com/scientificreports/

REA analysis using mean height (Sa)
We proposed the REA analysis method following the concept of the Representative Elementary Volume (REV) from continuum mechanics.REA is the 2D equivalent of REV in 3D analysis.REA is defined as the area when variations in Sa asymptote to a stead state as the AOI or 'L' is increased, as illustrated in Fig. 1a  steady-state Sa was denoted as Sa REA and its corresponding area as REA.Additional information on the calcula- tion of REA and steady-state Sa can be found in section "The REA analysis method" of the methods.Figures 6 and 7 illustrate Sa variations with AOI or length (L) with constant and variable y-axis limits, allowing a comparison of Sa behavior depending on polishing intensity.The increase in AOI length, L, showed a length-dependent variation in Sa for all polished quartz and glass surfaces until L exceeded 500 µm, leading to a steady state (Figs.6 and 7).These undulating variations in Sa indicated the absence of a fractal nature in the roughness resulting from polishing.
The magnitude of length-dependent Sa variations correlated with the polishing intensity, with coarser grit polishing (e.g., #60) exhibiting more significant Sa variations before reaching a steady state compared to finer grit polishing (e.g., #1200).Figures 6 and 7 illustrate steady-state Sa marked by red dashed lines, enabling the distinction of minor variations by considering two different y-axis limit modes (Figs.6a-f and 7a-f).Similarly, Figs. 6 and 7 illustrate the AOI length at steady-state Sa or REA marked by red diamonds.
Quartz surfaces showed no specific REA length dependence on polishing grit size, while glass surfaces exhibited a decrease in REA length as the polishing grit size became finer.This difference in REA was attributed to distinct surfaces generated from different polishing methods between quartz and glass surfaces.Notably, the REA for finely polished quartz and glass #1200 surfaces was substantially larger than single tile areas, and their corresponding Sa values significantly differed.When L < REA, Sa remained substantially unrepresentative.
This REA analysis demonstrated the need for a significantly larger surface area to determine a representative roughness parameter than is typically available through methods like AFM.It clarified why discrepancies may arise when determining roughness parameters, particularly when dealing with surfaces that cannot account for larger wavelength asperities due to their limited size.

Autocorrelation length, Sal, and REA
Whitehouse and Archard 33 proposed matching the sampling interval to the correlation length, which accommodates long-wavelength asperities.We postulate that REA may be associated with the correlation length,  offering insight into the estimation of representative roughness parameters.Here, we present an analysis of how the autocorrelation length (Sal) evolved by incrementally increasing the AOI to investigate its potential to reveal REA and roughness characteristics.Sal is a measure of the distance over which asperities exhibit correlation with a starting point, beyond which no correlation persists.This measure is defined as the horizontal span over which the autocorrelation function (ACF) decays to nil or 0.2.The ACF quantifies the correlation of a part of the surface concerning the entire AOI.The ACF is defined as a convolution of the surface with itself, shifted by ( τ x , τ y ), representing the spatial shift or 'lag' distance.It was computed as follows: Sal was then calculated from ACF as: For all polished quartz and glass surfaces, calculated Sal exhibited a linear increase in response to incremental AOI expansion, reaching a peak value, beyond which it decreased linearly (Fig. 8).Sal is known to characterize the wavelength structure of dominant asperity heights 39 , with smaller Sal values indicating surfaces dominated by high spatial frequency asperities, and vice versa.Thus, as the AOI expanded, Sal increased, signifying correlation over greater distances (i.e., L), until it reached a maximum correlation distance, as evident from the peaks in Fig. 8.While previous studies had reported a similar increase in Sal with sample size or AOI 25,39 , the subsequent decline in Sal beyond the peak warrants further investigation.One consideration can be that the linear increase of AOI only in the x-direction could create a biased shift of τ x relative to τ y , and secondly, Sal is the minimum of all correlation distances.
The peak Sal, or maximum autocorrelation length, denoted as Sal max , indicates the largest wavelength at which dominant asperity heights exhibit correlation or the minimum distance needed to identify all related asperities.Consequently, Sal max provides a reference to REA, albeit remaining significantly smaller than REA length (L).We found that Sal max followed an inverse power law with polishing grit # (Fig. 9) for glass surfaces, specifically of the form Sal max ∝ 1/grit# 1/3 (R 2 0.85).In contrast, no discernible trend was evident for quartz surfaces.However, it is worth noting that the REA length tended to be equal to or greater than the AOI length of Sal max .

Peaks and valleys
The polishing method exerted a significant influence on roughness characteristics, particularly concerning the extent and distribution of peaks and valleys (Figs. 3 and 4).Thus, we focused on understanding the influence of polishing on areal height parameters, Sp and Sv, representing the maximum peak height and the maximum valley      www.nature.com/scientificreports/for the examination of scale when the longest wavelength peaks and valleys were included in the evaluation, potentially revealing insights into REA and the impact of polishing on their relative magnitude.Peak and valley parameters, i.e., Sp and Sv, generally increased with increasing AOI length (L), eventually reaching a steady state asymptotically (Fig. 10).Initially, several intermediate steady states corresponding to multiscale roughness features could be observed.These intermediate states ultimately converged into a final steady state as AOI length expanded.The attainment of the final steady state signified that the occurrence of the highest peak and lowest valley of all wavelengths had been comprehensively incorporated within that length or AOI.
Notably, beyond the REA length, both Sp and Sv remained constant.However, steady values for Sp and Sv could be identified at lengths less than the AOI for both quartz and glass surfaces (Fig. 10).In Fig. 10, the vertical black dashed lines denote the REA.Furthermore, the initiation of the final steady-state Sp and Sv on numerous surfaces coincided with the REA length, underscoring the importance of encompassing the highest peaks and lowest valleys of all wavelengths in the determination of REA.
Valleys surpassed peaks in size (Fig. 10).As expected, the magnitude of the peak and valley decreased as the polishing fineness or grit # increased (Fig. 10).However, the magnitude of peaks on quartz surfaces decreased significantly with increasing polishing fineness (Fig. 10a-f).As a result, polishing could be considered as removing peaks selectively (Fig. 10g-l), leaving valleys as the dominant roughness characteristics.

Skewness and kurtosis
Skewness and kurtosis serve as height parameters that offer insights into the height distribution (z) of rough surfaces.Skewness, denoted as Sk, quantifies the symmetry of the height distribution within the surface topography.Positive Sk values indicate a prevalence of peaks, while negative values of Sk suggest a predominance of valleys.In instances where surface topography exhibits perfect symmetry and follows a Gaussian height distribution, Sk attains a value of zero.The calculation of Sk is expressed as follows:  Kurtosis ( κ ), on the other hand, serves to quantify the sharpness of the height distribution.It is a strictly positive value and indicates the extent of spikiness or bumpiness present.A high κ denotes a spiky surface, while a low κ characterizes a bumpy surface.For surfaces exhibiting a Gaussian height distribution, κ assumes a value of 3. The formula for calculating kurtosis (κ) is as follows: Skewness (Sk) and kurtosis (κ) were computed with a stepwise increase in AOI, thereby enabling an exploration of the length-scale dependency of these parameters and their potential in REA determination.The behavior of skewness (Sk) revealed a few undulations initially due to the multiscale nature of roughness on polished surfaces.It subsequently decreased as AOI increased, ultimately attaining steady negative values.This trend illustrated two significant observations: first, the length-scale dependency of Sk, and second, the prevalence of valleys as the dominant roughness feature on the studied surfaces.The magnitude of negative Sk intensified with the refinement of polishing, exemplifying how the stepwise finer polishing amplified the dominance of valleys as a roughness characteristic.Moreover, the AOI length at which Sk attained a steady state coincided with REA, thereby reinforcing REA analysis.In Fig. 11, the vertical black dashed lines denote the REA.
Conversely, kurtosis (κ), when studied with an incremental increase in AOI, exhibited a mirrored pattern in comparison to Sk. Initially displaying minor undulations, κ increased with increasing AOI, ultimately reaching steady-state positive values (Fig. 11).These steady κ values were > 3, suggesting a lognormal height distribution and the prevalence of spiky roughness, in contrast to bumpy roughness characteristics that are often associated with abrasive processes.Steady κ values grew with an increase in polishing fineness or grit size, thus indicating an augmentation in the spikiness feature for both quartz and glass surfaces.Additionally, the initiation of the final steady κ values aligned with REA, providing additional support for REA analysis alongside Sa (Fig. 11).

REA and Sa relations with polishing grit # and Sal max
Our analysis of data obtained from the roughness characterization of polished quartz and glass surfaces using parametric and functional methods prompts several key questions.We aim to investigate how the REA, essential for determining the representative mean height (Sa), is influenced by the polishing grit #.Additionally, we seek to discern any connections between REA and the maximum autocorrelation length, Sal max .Lastly, we delve into the relationship between the representative mean height, Sa, and polishing grit #.
In addressing these questions, we found that the smaller REA needed for finely polished glass surfaces led to an inverse power-law dependence between REA and polishing grit # (Fig. 12a), which took the form REA ∝ 1/grit# 1/4 (R 2 0.85).However, we observed no substantial trend between REA and the polishing grit # for quartz surfaces (Fig. 12b).This divergence arose from the distinctive polishing methods employed: sequential polishing for quartz and individual polishing for glass.Consequently, the quartz surfaces exhibited an indeterminate trend due to the formation of longer-wavelength valleys when each surface was sequentially polished from coarse to fine grit #.This led to the requirement of a relatively large REA even for finely polished quartz surfaces.
As noted in Sect.3.4, the REA length tended to be equal to or exceed the AOI length necessary to reach the maximum autocorrelation length, Sal max .Further examination revealed a linear relationship between L and Sal max (Fig. 12c,d).The glass surfaces exhibited a shallower slope in comparison to the quartz surface.In equation form, these relationships were represented as REA ∝ 1.4Sal max (R 2 0.97) for glass surfaces and REA ∝ 1.75Sal max (R 2 0.98) for quartz surfaces.These relationships demonstrated that the REA required for calculating representative Sa surpasses the Sal max , as suggested in prior research 25 , and this difference varied with the wavelength of asperities present on the surface.The presence of longer-wavelength valleys on quartz surfaces necessitated a relatively larger REA than the corresponding Sal max , reflected by the steeper slope of 1.75.Conversely, the glass surfaces, which did not feature longer-wavelength valleys due to the 'individual' polishing method, required a shorter REA than quartz surfaces, albeit still longer than the corresponding Sal max , as indicated by the gentler slope of 1.4 (Fig. 12).
It is expected that surface mean height, Sa, will depend on the degree of polishing.When considering the representative Sa, referred to as Sa REA , we observed an inverse power-law relationship between Sa REA and polish- ing grit size # (Fig. 13).These power-law models for glass and quartz surfaces were Sa REA ∝ grit# −0.55 (R 2 0.83) and Sa REA ∝ grit# −0.63 (R 2 0.66), respectively (Fig. 13).This variation in Sa REA was controlled by the material's hardness and crystallinity.Crystalline quartz has a hardness value of 7 on the Mohs scale, while amorphous glass has a hardness value of 5.5.Consequently, quartz exhibited more substantial peaks and valleys (as seen in Fig. 10), resulting in a larger Sa and a greater power-law exponent.

Sa uncertainty in the absence of REA analysis
REA analysis ensures the calculation of representative mean height, Sa.In the absence of REA analysis, the reported Sa values are susceptible to uncertainty.We were motivated to demonstrate how much uncertainty could be expected in Sa when REA analysis was not conducted.To achieve this, we computed both the maximum and minimum Sa values derived from the stepwise expansion of AOI and depicted them as error bars in Fig. 14.The red diamonds in the figure indicate the steady-state Sa or Sa REA .Additionally, we introduced Sa values obtained x, y dxdy from 1D line profiles taken at the top, middle, and bottom of 2D surfaces, allowing us to elucidate the disparity in uncertainty between 1 and 2D analyses.These Sa values from 1D profiles were represented by open blue circles, emphasizing the extent of potential variability (Fig. 14).
Our findings demonstrated that the extent of uncertainty, manifesting as variations in Sa, was magnified with coarser polishing grit sizes.Finely polished surfaces (grit # 1200) exhibited minimal Sa variability, typically deviating by 10% to 20% from Sa REA .In contrast, coarsely polished surfaces yielded considerable uncertainties.

Discussion and summary
Surface roughness determination of minerals is needed in various areas of geosciences and related engineering applications.Mineral surfaces play a pivotal role in processes such as sorption, precipitation-dissolution reactions, flow and transport phenomena, as well as multiphase saturation and transport through their influence on wettability.Despite this significance, the determination of mineral surface roughness has received inadequate attention.The most important challenge is how to determine a roughness parameter that is representative of all asperities found on a surface.Besides, extensive research demonstrates how the method, technique, or instrument used can influence roughness characterization.Attaining high-resolution surface roughness data is crucial for examining small-wavelength asperities.However, this often results in a limited scan area of 100 µm 2 or less.The limited scan areas cannot account for a wide range of wavelength asperities, rendering the assessment of roughness parameters contingent on the measurement scale or scan size 12,25,26,28,29,31 .When longer-wavelength asperities exist beyond the scan area, the determination of roughness becomes unrepresentative and inaccurate 12,25 .
This study aimed to establish surface roughness parameters that are representative of all asperities, which will promote reliable correlations between roughness parameters and their dependent phenomena, such as wettability 18 or boundary slip 44 .To achieve this, we proposed the REA analysis method following the concept of continuum mechanics.Quartz and glass surfaces were polished with various grit sizes, and Confocal Laser Scanning Microscopy was employed to combine multiple single scan tiles to obtain large scan areas up to 2500 µm in length.The study focused on the mean height (Sa) parameter and its convergence to a steady-state, which defined REA.
Our study revealed that even for finely polished surfaces, single tile scans measuring 129 µm × 96 µm were insufficient for determining a representative Sa.Attempts to deduce steady-state Sa or REA from single-tile scans by incrementally increasing the AOI led to erroneous steady-state Sa values (Fig. 2).This observation emphasizes the limitations of single-tile scans in capturing the complexity of surface roughness, even in precisely polished 'smooth' surfaces.To provide the sample area needed to include longer-wavelength asperities and thus determine a roughness parameter that is representative of all asperities found on a surface, multiple scan tiles must be combined.By combining surface data from up to ten scan tiles, our study unveiled a multiscale surface roughness texture influenced by polishing grit size and method.Coarser polishing introduced longer-wavelength asperities on both quartz and glass surfaces, while sequential coarse-to-fine polishing selectively removed peaks while preserving valleys on quartz surfaces.In contrast, individually polished glass surfaces exhibited a more even distribution of peaks and valleys.The stepwise finer polishing led to the selective elimination of peaks, with valleys emerging as the dominant roughness characteristics.As an example, this ratio of peaks to valleys is known to directly influence the Wenzel versus Cassie-Baxter state, controlling wettability characteristics.
Material hardness differences between glass and quartz significantly impacted surface roughness, with glass exhibiting roughly half the roughness of quartz due to its lower hardness.For example, crystalline quartz has a hardness value of 7 on the Mohs scale, while amorphous glass has a hardness value of 5.5.While the multiscale roughness texture observed might suggest a fractal nature of surface asperities, we found no evidence of a powerlaw relationship between the mean height (Sa) parameter and AOI length when applying the roughness-length method.
The novelty of this study lies in introducing the REA analysis method.We illustrated how REA analysis is required prior to determining representative roughness parameters.For example, this REA analysis method revealed undulating Sa variations for AOI lengths less than 500 µm, which converged to a steady state at lengths exceeding 500 µm for all surfaces.This highlights the necessity of conducting REA analysis before evaluating representative Sa.Besides, the persistence of Sa oscillations reinforces the absence of fractal roughness.Using the proposed method, we determined steady-state Sa (i.e., Sa REA ), which decreased with finer polishing grit, show- ing an inverse power-law relationship.Quartz required larger REA lengths due to persistent long-wavelength valleys induced by stepwise sequential polishing, while REA for steady-state Sa on glass decreased with finer grit, following an inverse power law.The REA analysis demonstrated that the surface area required to determine a representative roughness parameter is significantly greater than the area available, for instance when using AFM, explaining why the discrepancy in the determination of a roughness parameter could exist due to the use of length-limited surfaces that cannot take into account larger wavelength asperities 25 .
In addition to steady-state Sa, our study noted a convergence to a steady state in various parametric and functional roughness parameters, such as peaks (Sp), valleys (Sv), skewness (Sk), kurtosis (κ), and autocorrelation length (Sal).Sp and Sv displayed an asymptotic increase with AOI length, reaching a steady state at lengths analogous to REA.Sk exhibited a decline to steady-state negative values with increasing AOI length, underscoring the prevalence of valleys as prominent roughness features.The behavior of kurtosis (κ) mirrored that of Sk.A steady state κ of > 3 indicated the prevalence of spiky roughness, in contrast to bumpy roughness characteristics that are known to result from abrasive processes.The autocorrelation length (Sal) exhibited linear increases leading to a peak value before decreasing linearly with incremental AOI length.Although prior studies have reported similar increases in Sal with sample size or AOI 25,39 , the observed reduction in Sal beyond the peak value remains a topic of further investigation.We found no correlation between Sal/L and REA, which contradicts the proposition by Nečas et al., 25 that a Sal/L ratio < 0.1 indicates reduced bias and sample size length for obtaining representative roughness.The maximum or peak Sal, denoted as Sal max , signifies the largest wavelength of correlated asperity heights or the smallest distance required to include all pertinent asperities.We found a linear relationship between Sal max and REA, with a steeper slope of 1.75 for quartz and a gentler slope of 1.4 for glass surfaces.Thus, Sal max clearly offered a reference to REA, although it remained consistently smaller, indicating that the REA required for determining the representative Sa can be larger than the maximum Sal 25 , depending on roughness characteristics caused by material hardness and possibly polishing method.
In the absence of the proposed REA analysis, uncertainty in reported Sa can thus be expected.We found that the magnitude of uncertainty depends on the polishing grit size.Finely polished surfaces displayed a smaller variability of 10%-20% relative to steady-state Sa, which got amplified to up to 70% of steady-state Sa with coarser polishing.Despite using significantly longer 1D profiles from combined scan tiles, the Sa from these 1D profiles remained inadequate because they underestimated the peaks and valleys, resulting in smaller Sa.Therefore, we emphasize on the significance of conducting the proposed REA analysis prior to calculating representative surface roughness parameters from 2D profiles.This proposed novel method will facilitate reliable correlations of roughness parameters with physiochemical phenomena, ultimately advancing our understanding and control of processes influenced by surface roughness in geosciences and related engineering applications.

Figure 1 .
Figure 1.Single tile scans comparing surface texture between the finest polished (a) quartz and (b) glass surfaces and engineered (c) frosted glass and (d) clear glass surfaces.

Figure 2 .Figure 3 .
Figure 2. Mean height, Sa variation with sample length, L within single scan tiles shown in Fig. 1.

Figure 4 .
Figure 4.A linear combination of tile scans showing longer-wavelength surface texture variation associated with differences in the magnitude of polishing (a-f) on glass surfaces.

Figure 6 .
Figure 6.Quartz surface Sa variation with length, L. Steady state Sa is denoted by red dashed lines.REA is marked by red diamonds.Figures to the left(i.e., a to f) use a constant y-axis for comparison, whereas figures on the right (i.e., g to l) focus on Sa variation with variable y-axis limits.

(Figure 7 .
Figure 7. Glass surface Sa variation with length, L. Steady state Sa is denoted by red dashed lines.REA is marked by red diamonds.Figures to the left(i.e., a-f) use a constant y-axis for comparison, whereas figures on the right (i.e., g-l) focus on Sa variation with variable y-axis limits.

Figure 8 .
Figure 8. Autocorrelation length (Sal) variation with a stepwise increase in the sample window or AOI for polished quartz (a-f) and glass (g-l) surfaces.Black dashed lines denote the REA length.

Figure 9 .
Figure 9.The dependence of the maximum autocorrelation length, Sal max on polishing grit size #.

Figure 11 .
Figure 11.Variation in Sk and κ for polished quartz with a stepwise increase in the sample window or AOI of quartz (a-f) and glass (g-l) surfaces.Black dashed lines denote the REA length.

Figure 13 .
Figure 13.The dependence of steady-state mean height, Sa REA on polishing grit size #.

Figure 14 .
Figure 14.Uncertainty in surface roughness, Sa dependent on the magnitude of roughness or polishing grit size # and 1D vs. 2D analysis method.

Table 1 .
. Notations for parameters and abbreviations used.*In accordance with ISO 25178 1 standard.
For instance, quartz polished with grit # 80 registered a 2D Sa of 0.56 µm and a 1D Sa of 0.49 µm, whereas Sa REA stood at 1.77 µm.Similarly, glass surfaces polished with the coarsest grit sizes presented substantial Sa variability.Notably, quartz surfaces displayed greater Sa variability compared to glass surfaces, potentially stemming from the variation in longer-wavelength deeper valleys due to distinct polishing methods.Furthermore, the Sa values derived from 1D profiles, denoted as Sa 1D , consistently fell below Sa REA , primarily due to the underestimation of peaks and valleys inherent to the 1D profiles.